text: Consider quantum chromodynamics (QCD) with two massless quarks  and  (massive fermions do not exhibit chiral symmetry). The Lagrangian reads
  In terms of left-handed and right-handed spinors, it reads
  (Here,  is the imaginary unit and  the Dirac operator.)
  Defining
  it can be written as
  The Lagrangian is unchanged under a rotation of q by any 2×2 unitary matrix , and q by any 2×2 unitary matrix .
  This symmetry of the Lagrangian is called flavor chiral symmetry, and denoted as .   It decomposes into
  The singlet vector symmetry, ,  acts as 
  and thus invariant under  gauge symmetry. This corresponds to baryon number conservation.
  The singlet axial group    transforms as the following global transformation
  However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by a quantum anomaly.
  The remaining chiral symmetry  turns out to be spontaneously broken by a quark condensate 
   formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup  known as isospin. The Goldstone bosons corresponding to the three broken generators are the three pions. 
  As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this chiral symmetry breaking induces the bulk of hadron masses, such as those for the nucleons &mdash; in effect, the bulk of the mass of all visible matter.
  In the real world, because of the nonvanishing and differing masses of the quarks,  is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.